Optimal fault-tolerant implementations of Heisenberg interactions and controlled-Za gates

ABSTRACT

The disclosure describes various aspects of techniques for optimal fault-tolerant implementations of controlled-Z a  gates and Heisenberg interactions. Improvements in the implementation of the controlled-Z a  gate can be made by using a clean ancilla and in-circuit measurement. Various examples are described that depend on whether the implementation is with or without measurement and feedforward. The implementation of the Heisenberg interaction can leverage the improved controlled-Z a  gate implementation. These implementations can cut down significantly the implementation costs associated with fault-tolerant quantum computing systems.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application claims priority to and the benefit of U.S.Provisional Patent Application No. 62/632,767, entitled “OPTIMALFAULT-TOLERANT IMPLEMENTATIONS OF HEISENBERG INTERACTIONS ANDCONTROLLED-Z^(a) GATES,” and filed on Feb. 20, 2018, the contents ofwhich are incorporated herein by reference in their entirety.

BACKGROUND OF THE DISCLOSURE

Aspects of the present disclosure generally relate to quantum gates, andmore specifically, to techniques for optimizing the implementation ofquantum gates in fault-tolerant quantum computing systems.

Trapped atomic ions and superconducting circuits are two examples ofquantum information processing (QIP) approaches that have delivereduniversal and fully programmable machines. In these types of quantuminformation processing or quantum computing systems, it is important toprotect the quantum information from errors that may result from, forexample, decoherence and other quantum noise. As such, fault-tolerantquantum computing systems employ quantum error correction techniques todeal not only with the noise that may exist in the quantum information,but also with faults in the quantum gates, the quantum statepreparation, and the measurements in the systems.

A controlled-Z^(a) gate (that is, a gate that provides a Z rotation witharbitrary parameter a) is a type of quantum gate that is widely used inquantum computing. For example, the controlled-Z^(a) gate is used in aversatile quantum algorithmic subroutine called the quantum Fouriertransform (QFT). The implementation of the controlled-Z^(a) gate infault-tolerant quantum computing systems, however, may be quiteexpensive in terms of different types of resources (including physicalresources, operating time). In addition, the so-called Heisenberginteraction, which may be used in digital-level quantum simulations, mayalso be quite expensive to implement. As such, it may be desirable tohave more efficient or less costly implementations of thecontrolled-Z^(a) gate and the Heisenberg interaction for an optimalcircuit compilation of quantum algorithms and quantum simulations.

SUMMARY OF THE DISCLOSURE

The following presents a simplified summary of one or more aspects inorder to provide a basic understanding of such aspects. This summary isnot an extensive overview of all contemplated aspects, and is intendedto neither identify key or critical elements of all aspects nordelineate the scope of any or all aspects. Its purpose is to presentsome concepts of one or more aspects in a simplified form as a preludeto the more detailed description that is presented later.

This disclosure describes improved ways to implement a controlled-Z^(a)gate using a clean ancilla and an in-circuit measurement. Thisdisclosure also shows an optimized implementation of the Heisenberginteraction that makes use of the improved controlled-Z^(a) gateimplementation. In both instances, the cost of implementingfault-tolerant quantum computations is significantly reduced.

In an aspect, a method for performing a quantum algorithm is describedthat includes identifying use of a controlled-Z^(a) gate as part of thequantum algorithm, wherein a is a parameter and a∈[−1, 1]; implementingthe controlled-Z^(a) gate for a fault-tolerant QIP system, wherein theimplementation of the controlled-Z^(a) gate includes multiple elementswith only six (6) of the elements being controlled-NOT (CNOT) gates;mapping the implementation of the controlled-Z^(a) gate into a physicalrepresentation in the fault-tolerant QIP system; and performing thequantum algorithm based at least in part on the physical representation.

In another aspect, a different method for performing a quantum algorithmis described that includes identifying use of a controlled-Z^(a) gate aspart of the quantum algorithm, wherein a is a parameter and a∈[−1, 1];implementing the controlled-Z^(a) gate for a fault-tolerant QIP system,wherein the implementation of the controlled-Z^(a) gate includesmultiple elements with only four (4) of the elements being T gates, onlythree (3) of the elements being CNOT gates, and only three (3) of theelements being Hadamard (H) gates; mapping the implementation of thecontrolled-Z^(a) gate into a physical representation in thefault-tolerant QIP system; and performing the quantum algorithm based atleast in part on the physical representation.

In yet another aspect, a method for performing a quantum simulation isdescribed that includes identifying use of a Heisenberg interaction aspart of the quantum simulation; identifying a controlled-Z^(a) gate forimplementing the Heisenberg interaction, wherein a is a parameter anda∈[−1, 1]; implementing the Heisenberg interaction for a fault-tolerantQIP system, wherein the implementation of the Heisenberg interaction isbased on an implementation of the controlled-Z^(a) gate and includesmultiple elements with only one (1) of the elements being a parametrizedZ^(a) gate; mapping the implementation of the Heisenberg interactioninto a physical representation in the fault-tolerant QIP system;performing the quantum simulation based at least in part on the physicalrepresentation; and providing results from the quantum simulation.

Also described herein are apparatuses and computer-readable storagemedium for various methods described above associated with optimalfault-tolerant implementations of the Heisenberg interaction andcontrolled-Z^(a) gates.

BRIEF DESCRIPTION OF THE DRAWINGS

The appended drawings illustrate only some implementation and aretherefore not to be considered limiting of scope.

FIG. 1 is a diagram that illustrates an example of a definition of arelative phase (R) Toffoli gate in accordance with aspects of thisdisclosure.

FIG. 2 is a diagram that illustrates an example of an ancilla-aidedfault-tolerant controlled-Z^(a) gate in accordance with aspects of thisdisclosure.

FIG. 3 is a diagram that illustrates an example of an ancilla-aided,in-circuit measurement-based fault-tolerant controlled-Z^(a) gate inaccordance with aspects of this disclosure.

FIG. 4 is a diagram that illustrates an example of a quantum circuit fora two-body interaction with an arbitrary parameter θ in accordance withaspects of this disclosure.

FIG. 5 is a diagram that illustrates an example of an implementation ofthe Heisenberg interaction optimal in the number of real-valued degreesof freedom up to a global phase on e^(−ia) in accordance with aspects ofthis disclosure.

FIG. 6 is a diagram that illustrates an example of the CNOT-countoptimal implementation of the Heisenberg interaction up to a globalphase on e^(−iπ/4) in accordance with aspects of this disclosure.

FIG. 7 is a diagram that illustrates an example of a computer device inaccordance with aspects of this disclosure.

FIG. 8 is a flow diagram that illustrates an example of a method forperforming a quantum algorithm in accordance with aspects of thisdisclosure.

FIG. 9 is a flow diagram that illustrates an example of a method forperforming a quantum simulation in accordance with aspects of thisdisclosure.

FIG. 10 is a block diagram that illustrates an example of atrapped-ion-based QIP system in accordance with aspects of thisdisclosure.

DETAILED DESCRIPTION

The detailed description set forth below in connection with the appendeddrawings is intended as a description of various configurations and isnot intended to represent the only configurations in which the conceptsdescribed herein may be practiced. The detailed description includesspecific details for the purpose of providing a thorough understandingof various concepts. However, it will be apparent to those skilled inthe art that these concepts may be practiced without these specificdetails. In some instances, well known components are shown in blockdiagram form in order to avoid obscuring such concepts.

It is to be understood that various aspects of this disclosure relate tofault-tolerant quantum computing systems. In physical quantum computingsystems, at the physical level, there are likely to be some errors inquantum computations. Therefore, fault-tolerant quantum computingsystems may be used that implement techniques such as performing quantumcomputations multiple times to identify the ones that are error free andproceed based on those quantum computations. The implementation ormapping of quantum gates or quantum operations (e.g., computing or gateprimitives) from a logical expression or level (e.g., encoded logicallevel gates) to the physical level come with some costs forfault-tolerant quantum computing systems depending on the type offault-tolerant protocols used. These costs can include the overallnumber of physical qubits that are needed and/or the number ofoperations that need to be performed as part of a computation, in effectthe time to complete the computation. For some gates, such as Cliffordgates, it is generally known how to implement them over arbitraryphysical level quantum computers (being trapped-ion, superconducting, orother type of quantum computes). Such gates as well as other types ofgates can be expensive to implement but are generally needed in manyimportant quantum computations. For example, for most fault-tolerantprotocols, the implementation of T gates is very costly.

As described above, the controlled-Z^(a) gate is widely used in quantumcomputing. For example, the controlled-Z^(a) gate is used in the QFT.The controlled-Z^(a) gate is an important computing primitive and is notdirectly implementable in a fault-tolerant quantum computing system.When implementing the controlled-Z^(a) gate in fault-tolerant quantumcomputing systems, however, such implementation may be quite expensivein terms of different types of resources. In addition, the Heisenberginteraction may also be quite expensive to implement. As such, it may bedesirable to have more efficient or less costly implementations of thecontrolled-Z^(a) gate and the Heisenberg interaction for an optimalcircuit compilation of quantum algorithms and quantum simulations.

The present disclosure describes an improved fault-tolerantimplementation of the widely used controlled-Z^(a) gate over Clifford+Tlibrary, with or without the availability of in-circuit measurement andfeedforward. The present disclosure also shows an optimizedfault-tolerant implementation of the so-called Heisenberg interaction.This is accomplished by using a single controlled-Z^(a) rotation gate toaddress the real-valued degree of freedom in the Heisenberg interaction,as a whole. In comparison to other implementations, the implementationsdescribed in this disclosure can improve the fault-tolerant cost of thecontrolled-Z^(a) gate from, for example, eight (8) T gates and oneparametrized Z^(a) gate (described in more detail below) to four (4) Tgates and one parametrized Z^(a) gate. Relying at least partially onthese improvements, the fault-tolerant cost of the Heisenberginteraction may also be improved from, for example, three (3)parametrized Z^(a) gates to one parametrized Z^(a) gate and four (4) Tgates, which is almost a factor of three.

Because the Heisenberg interactions are frequently used in quantumsimulations of condensed matter systems and controlled-Z^(a) gatesconstitute one of the basic building blocks of the quantum Fouriertransform, which is used in the phase estimation and polynomial-timediscrete logarithm algorithms for Abelian groups (including the originalShor's integer factoring), the optimization results described in thisdisclosure may be applicable to a wide range of quantum algorithms.

Since, typically, Clifford gates are less costly to implement and Tgates are more expensive to implement in fault-tolerant quantumcomputing systems, this disclosure focuses primarily on techniques forreducing the count or number of T gates. For a Z^(a) gate, which is notamenable to direct fault-tolerant implementation, the cost ofimplementing the Z^(a) gate may change based on the different ways ofapproximating it. In fact, known ways to approximate a given(parametrized) Z^(a) gate to the error e require logarithmically (in1/ε) many T gates, whether the approximation is done optimally using asingle qubit over unitary circuits, or by employing additional resourcesin the form of ancilla (e.g., an ancilla qubit), measurement, andfeedforward. Moreover, the quality of the approximation of Z^(a) neednot depend on the value of a (so long as it is not an integer multipleof ¼), but rather on the desired approximation error ε.

Described below are techniques on how to synthesize a fault-tolerantimplementation of the controlled-Z^(a) gate. In particular, thefault-tolerant implementation described herein requires a clean ancillaqubit. Efficient construction of such a gate is needed in realizing, forexample, the widely used quantum Fourier transform (QFT), or as it willbe described further below, the Heisenberg interaction.

To begin, the parametrized gate Z^(a), where a∈[−1, 1], and theparametrized gate R_(z)(θ), where θ∈[0, 2π], may be defined as describedin equation (1):

$\begin{matrix}{{Z^{a}:=\begin{pmatrix}1 & 0 \\0 & e^{i\;\pi\; a}\end{pmatrix}},{{R_{z}(\theta)}:=\begin{pmatrix}e^{{- i}\;{\theta/2}} & 0 \\0 & e^{i\;{\theta/2}}\end{pmatrix}},} & (1)\end{matrix}$where a∈[−1, 1] means that the value of a is in the range between −1 and1, including the end values of the range, that is, −1 and 1, and whereθ∈[0, 2π] means that the value of θ is in the range between 0 and 2π,including the end values of the range, that is, 0 and 2π. With such adefinition, Z¹=Z (for a=1) is the well-known Pauli-Z gate, Z^(1/2)=P(for a=½) is the Phase gate, Z^(−1/2)=P^(†) (for a=−½) is the complexconjugate of the Phase gate, Z^(1/4)=T (for a=¼) is the T gate, andZ^(−1/4)=T^(†) (for a=−¼) is the complex conjugate of the T gate. It isalso noted that because of the difference between Z^(a) and R_(z)(θ) bya global phase (for the proper choice of a and θ), fault-tolerantimplementations other than the ones discussed herein may apply to theR_(z)(θ) gates as well.

In the various techniques described herein, the circuit implementationof the quantum simulation algorithm whose simulation dynamics consistsin part of the Heisenberg interaction may be laid out such that quantumgate optimization opportunities are exposed as much as possible.Specifically, the simulation circuit based on a product formulaalgorithm may lay out the individual Heisenberg interaction terms inalternate orders—forward and reverse—to ensure that a maximumoptimization is obtained from the application of a circuit optimizer.This may be performed as part of the pre-preprocessing of the circuitoptimizer in the case of any Trotterization-based quantum simulationinput. The circuit optimizer may be configured to natively handlecontrolled-Z^(a) gates and may apply the merging rule controlled-Z^(a)controlled-Z^(b)=controlled-Z^((a+b)). This is a straightforwardextension of the functionality of the circuit optimizer to accommodate acontrolled version of the R_(z) gate. Another feature that may beimplemented is the merging of two Heisenberg interactions as anoptimization rule. This functionality may fit best into thepre-processing of, for example, a circuit optimizer so as to immediatelydiscover and merge two Heisenberg interactions into one before anyoptimization is done. Yet another feature that may be implemented is thecapability of being able to handle classically controlled quantum gates.The classically controlled part of the gate is a pure quantum gate. Theclassical control may or may not be realized based on in-circuitmeasurements to be done in an actual instance of quantum computation. Aslong as these two possible scenarios are compatible with any other partof the circuit optimizer, it is possible to straightforwardlyaccommodate for classically controlled gates. For example, whenconsidering commutation of two gates to the preceding and following of aclassically controlled gate, commutability may be separately checked forthe two cases: (i) the pure quantum part is applied, then use the rulesthat are already built into the circuit optimizer, and (ii) the purequantum part is not applied, trivially commute things through since nogate is applied.

FIG. 1 shows a diagram 100 that illustrates a definition of a relativephase Toffoli gate (R) used in this disclosure having multiple elements,where H is a Hadamard gate (H 110), T is a T gate (T 120), T^(†) is thecomplex conjugate of the T gate (T^(†) 130), and the symbol ⊕ connectingto a solid (dark) circle corresponds to a CNOT gate or controlled-NOTgate (CNOT 140).

FIG. 2 shows a diagram 200 that illustrates an ancilla-aidedfault-tolerant controlled-Z^(a) gate having multiple elements. In thisimplementation, by using one ancilla (e.g., an ancilla qubit), it ispossible to obtain a fault-tolerant gate count cost of eight (8) T gatesand a Z^(a) gate (Z^(a) 210). For example, based on the definition ofthe relative phase Toffoli gate in FIG. 1 , the diagram 200 in FIG. 2effectively includes one relative phase Toffoli gate (2 T gates and 2T^(†) gates) to the left of the Z^(a) 210 and another relative phaseToffoli gate (2 T gates and 2 T^(†) gates) to the right of the Z^(a)210. Moreover, the ancilla qubit in this example is set to 10).

The implementation shown in FIG. 2 may have the same fault-tolerant gatecount as other reported implementations where a middle T gate may bereplaced by a Z^(a) gate. The implementation shown in FIG. 2 , however,may use only six (6) CNOT gates (3 for each relative phase Toffoli gate)in contrast to twelve (12) CNOT gates used in these otherimplementations, and does not use any Phase gates, where these otherimplementations use two (2) Phase gates. That is, even though the gatecount may be the same, the resources needed (e.g., the cost ofimplementing the gates) is much less in the implementation shown in FIG.2 compared to other implementations.

FIG. 3 shows a diagram 300 that illustrates an ancilla-aided, in-circuitmeasurement-based fault-tolerant controlled-Z^(a) gate having multipleelements. In this implementation, by using one ancilla (e.g., an ancillaqubit), in-circuit measurement 320, and feedforward (e.g., using aclassically-conditioned controlled Pauli-Z gate), it is possible toobtain a fault-tolerant gate count cost of four (4) T gates and an R_(z)gate. For example, based on the definition of the relative phase Toffoligate in FIG. 1 , the diagram 300 in FIG. 3 effectively includes onerelative phase Toffoli gate (2 T gates and 2 T^(†) gates) to the left ofa Z^(a−1/4) gate (Z^(a−1/4) 310), in addition to a Hadamard gate (H110), the in-circuit measurement 320, and a classically-conditionedcontrolled-Z gate (Z 330). Moreover, the ancilla qubit in this exampleis initialized to |0).

The implementation shown in FIG. 3 is based at least in part oncombining Kitaev's trick with Toffoli-measurement construction, over anefficient relative phase Toffoli gate, and other circuitsimplifications.

In addition to the fault-tolerant implementations of thecontrolled-Z^(a) gate described above in connection with FIGS. 1-3 , thedisclosure also describes aspects of the Heisenberg interaction inquantum simulations. For example, typically in digital-level quantumsimulations, a given local Hamiltonian term (an addend of the entiresystem Hamiltonian) may be projected onto a well-known set of Paulibases, of which there are 2^(k), where k is the number of qubits thatparticipate in simulating the given local Hamiltonian term. For variousreasons, including many body localization phenomena in condensed matter,the so-called Heisenberg interaction plays an important role in suchsimulations.

A standard or known approach to implement the Heisenberg interactionrequires 6 CNOT gates, in addition to three R_(z) gates. That is, eachthree two-body interactions {circumflex over (σ)}_(x){circumflex over(σ)}_(x), {circumflex over (σ)}_(y){circumflex over (σ)}_(y), and{circumflex over (σ)}_(z){circumflex over (σ)}_(z) is implemented usingone R_(z) gate and two (2) CNOT gates.

An improved implementation of the Heisenberg interaction is describedbelow. This implementation reduces the pre-fault-tolerant cost to justfour (4) CNOT gates, while projecting the real-valued degree of freedomin the Heisenberg interaction onto the controlled-R_(z) rotation. Assuch, this proposed implementation is optimal in the real-valued degreesof freedom, and furthermore is well suited for implementation overfault-tolerant quantum computing systems or machine because of mappingof the existing real-valued degree of freedom into that of thesingle-qubit Z^(a) gate through application of the controlled-Z^(a) gateimplementations described above in connection with FIGS. 2 and 3 .

The Hamiltonian for the Heisenberg interaction between bodies labeled iand j takes the form Ĥ_(ij)={right arrow over (σ)}^((i))·{right arrowover (σ)}^((j)), where {right arrow over (σ)}^((i))=({circumflex over(σ)}_(x) ^((i)),{circumflex over (σ)}_(y) ^((i)),{circumflex over(σ)}_(z) ^((i)))^(T). Because this Hamiltonian includes the addition ofthree Pauli bases, that is, {circumflex over (σ)}_(x){circumflex over(σ)}_(x), {circumflex over (σ)}_(y){circumflex over (σ)}_(y), and{circumflex over (σ)}_(z){circumflex over (σ)}_(z), where the i and jindices are dropped hereinafter for simplicity, the standard Trotterformula approach to simulating such a Hamiltonian on a quantum computerwould then require implementations of each of the three Pauli productterms. In an example of the {circumflex over (σ)}_(z){circumflex over(σ)}_(z) interaction, the evolution operator readse^(−i{circumflex over (σ)}) ^(z) ^({circumflex over (σ)}) ^(z) ^(θ) andmay be implemented as a quantum circuit with some arbitrary parameter asdescribed in a diagram 400 shown in FIG. 4 , which includes two (2) CNOTgates (CNOT 140) and an R_(z)(2θ) gate (R_(z)(2θ) 410).

Circuits for the {circumflex over (σ)}_(x){circumflex over (σ)}_(x)interaction and the {circumflex over (σ)}_(y){circumflex over (σ)}_(y)interaction may be similarly constructed by using the example quantumcircuit shown in FIG. 4 and applying a basis change.

Because in generale^(−i({circumflex over (X)}+Ŷ))≠e^(−i({circumflex over (X)}))e^(−i(Ŷ))for {circumflex over (X)} and Ŷ matrices, the Trotterized implementationof e^(−i{circumflex over (σ)}) ^(x) ^({circumflex over (σ)}) ^(x) ^(θ)followed by e^(−i{circumflex over (σ)}) ^(y) ^({circumflex over (σ)})^(y) ^(θ) and e^(−i{circumflex over (σ)}) ^(z) ^({circumflex over (σ)})^(z) ^(θ) on a quantum computer or quantum computing system is not ingeneral exactly the same ase^(−i{right arrow over (σ)}·{right arrow over (σ)}θ), which is theoriginal local evolution that one targeted to implement.

A Trotter formula-based quantum simulation that includes Heisenberginteractions would therefore suffer from the loss of accuracy in thestandard implementation approach because of the approximate nature ofthe implementation as described above. This means that in order to meeta predetermined error tolerance level of the full simulation, additionalquantum computational efforts need be expended. Accordingly, a betterquality implementation of the Heisenberg interaction may save the needto expend additional quantum resources since the simulation, and theresults from those simulations, would no longer suffer from the loss ofaccuracy that arises from the poor quality implementation.

To address these issues, this disclosure proposes a different approachto implement the Heisenberg interaction. This implementation is shown ina diagram 500 in FIG. 5 and is represented by the definition describedin equation (2) below:

$\begin{matrix}{{{Heisenberg}(a)}:={e^{{- {i{({{{\hat{\sigma}}_{x}{\hat{\sigma}}_{x}} + {{\hat{\sigma}}_{y}{\hat{\sigma}}_{y}} + {{\hat{\sigma}}_{z}{\hat{\sigma}}_{z}}})}}}a} = {\begin{pmatrix}e^{- {ia}} & 0 & 0 & 0 \\0 & {e^{- {ia}}{\cos\left( {2a} \right)}} & {e^{- {i{({a - {\pi/2}})}}}{\sin\left( {2a} \right)}} & 0 \\0 & {e^{- {i{({a - {\pi/2}})}}}{\sin\left( {2a} \right)}} & {e^{- {ia}}{\cos\left( {2a} \right)}} & 0 \\0 & 0 & 0 & e^{- {ia}}\end{pmatrix}.}}} & (2)\end{matrix}$

This implementation may require a single Z^(a) rotation, whileimplementing the Heisenberg interaction exactly, because thecontrolled-Z^(a) gate sub-circuit in FIG. 5 (Z^(4a) 510) is implementedas described above in connection with FIGS. 2 and 3 . The implementationof the Heisenberg interaction shown in FIG. 5 is optimal in the numberof real-valued degrees of freedom, up to a global phase of e^(−ia). Asmentioned above, this implementation of the Heisenberg interaction canbe made with the controlled-Z^(a) gates as implemented in FIGS. 2 and 3depending on whether fault-tolerant implementation with or withoutmeasurement and feedforward is preferred. The implementation of theHeisenberg interaction in FIG. 5 also includes two (2) Hadamard gates (H110) and two (2) CNOT gates (CNOT 140).

If the focus of the implementation of the Heisenberg interaction is onthe pre-fault-tolerant cost, and as such consider the task of minimizingthe number of two-qubit interactions (CNOT gates) used in theimplementation, then it is possible to implement the Heisenberginteraction using only three (3) CNOT gates. FIG. 6 shows a diagram 600that illustrates the corresponding circuit implementation of theHeisenberg interaction. This implementation includes three (3) R_(z)gates (e.g., R_(z)(−π/2) 610, R_(z)(2a−π/2) 630, and R_(z)(π/2) 650) andtwo (2) R_(y) gates (e.g., R_(y)(π/2−2a) 620 and R_(y)(2a−π/2) 640).

As described above, the disclosure provides two differentimplementations of controlled-Z^(a) gates regardless of the parameter abeing used. Moreover, the disclosure provides two differentimplementations of the Heisenberg interaction, each based on acorresponding controlled-Z^(a) gate implementation.

For example, a first implementation of the controlled-Z^(a) gate usesone (1) Z^(a) gate, eight (8) T gates, six (6) CNOT gates, four (4)Hadamard gates, and one (1) ancilla, as illustrated by expanding thediagram 200 in FIG. 2 . The best known implementation of thecontrolled-Z^(a) gate relies on one (1) Z^(a) gate, eight (8) T gates,twelve (12) CNOT gates, four (4) Hadamard gates, two (2) P gates, andone (1) ancilla.

The first implementation of the controlled-Z^(a) provides a costreduction over existing implementations. For example, based on costsinvolving at least a number of physical qubits and/or the number/timeinvolved in performing a computation, a cost of implementing a Z^(a)gate is approximately 20-50 times a cost of implementing a T gate, andas mentioned above, T gates are expensive gates to implement. A CNOTgate is approximately 50 times less expensive to implement than a Tgate. Hadamard gates, ancilla qubits, measurements, andclassically-conditioned controlled Pauli-Z gates are not very expensive,each is approximately the cost of implementing a CNOT gate, perhaps evencheaper. In view of this, the cost reduction of the first implementationcan be considered to be somewhat small.

In another example, a second implementation of the controlled-Z^(a) gate(with measurement and feedforward) uses one (1) Z^(a) gate, four (4) Tgates, three (3) CNOT gates, three (3) Hadamard gates, one (1) ancilla,one (1) measurement, and one (1) classically-conditioned controlledPauli-Z gate, as illustrated by expanding the diagram 300 in FIG. 3 .The best known implementation of the controlled-Z^(a) gate relies on one(1) Z^(a) gate, eight (8) T gates, twelve (12) CNOT gates, four (4)Hadamard gates, two (2) P gates, and one ancilla. The secondimplementation of the controlled-Z^(a) provides a reduction of four (4)T gates over existing implementations, where T gates are expensive toimplement.

In another example, a first implementation of the Heisenberg interactionuses one (1) Z^(a) gate, eight (8) T gates, eight (8) CNOT gates, six(6) Hadamard gates, and one (1) ancilla, as illustrated by expanding thediagram 500 in FIG. 5 , where the controlled-Z^(a) gate is implementedbased on the configuration shown in FIG. 2 . A standard (non-exact)implementation of the Heisenberg interaction relies on three (3) Z^(a)gates, six (6) CNOT gates, eight (8) Hadamard gates, two (2) P gates,and two (2) P^(†) gates. The implementation costs of the firstimplementation of the Heisenberg interaction are about three (3) timesbetter than the costs associated with the standard implementationbecause two (2) fewer Z^(a) gates are needed, and Z^(a) gates are evenmore expensive to implement than T gates.

In yet another example, a second implementation of the Heisenberginteraction uses one (1) Z^(a) gate, four (4) T gates, five (5) CNOTgates, five (5) Hadamard gates, one (1) ancilla, one (1) measurement,and one (1) classically-conditioned controlled Pauli-Z, as illustratedby expanding the diagram 500 in FIG. 5 , where the controlled-Z^(a) gateis implemented based on the configuration shown in FIG. 3 . A standard(non-exact) implementation of the Heisenberg interaction relies on three(3) Z^(a) gates, six (6) CNOT gates, eight (8) Hadamard gates, two (2) Pgates, and two (2) P^(†) gates. The implementation costs of the secondimplementation of the Heisenberg interaction are about three (3) timesbetter than the costs associated with the standard implementationbecause two (2) fewer Z^(a) gates are needed, and Z^(a) gates are evenmore expensive to implement than T gates.

Referring now to FIG. 7 , illustrated is an example computer device 700in accordance with aspects of the disclosure. The computer device 700can represent a single computing device, multiple computing devices, ora distributed computing system, for example. The computer device 700 maybe configured as a quantum computer, a classical computer, or acombination of quantum and classical computing functions. For example,the computer device 700 may be used to process information to generateor determine the various quantum circuit constructions orimplementations described herein that use a controlled-Z^(a) gate and/ora Heisenberg interaction. Moreover, the computer device 700 may be usedas a quantum computer and may implement quantum algorithms and/orquantum simulations based on the quantum circuit constructions orimplementations described herein. A generic example of a quantuminformation processing (QIP) system that can implement and use thequantum circuit constructions or implementations described herein isillustrated in an example shown in FIG. 10 .

In one example, the computer device 700 may include a processor 710 forcarrying out processing functions associated with one or more of thefeatures described herein. The processor 710 may include a single ormultiple set of processors or multi-core processors. Moreover, theprocessor 710 may be implemented as an integrated processing systemand/or a distributed processing system. The processor 710 may include acentral processing unit (CPU), a quantum processing unit (QPU), agraphical processing unit (GPU), or combination of those types ofprocessors.

In an example, the computer device 700 may include a memory 720 forstoring instructions executable by the processor 710 for carrying outthe functions described herein. In an implementation, for example, thememory 720 may correspond to a computer-readable storage medium thatstores code or instructions to perform one or more of the functions oroperations described herein. In one example, the memory 720 may includeinstructions to perform aspects of a method 800 and a method 900described below in connection with FIGS. 8 and 9 .

Further, the computer device 700 may include a communications component730 that provides for establishing and maintaining communications withone or more parties utilizing hardware, software, and services asdescribed herein. The communications component 730 may carrycommunications between components on the computer device 700, as well asbetween the computer device 700 and external devices, such as deviceslocated across a communications network and/or devices serially orlocally connected to computer device 700. For example, thecommunications component 730 may include one or more buses, and mayfurther include transmit chain components and receive chain componentsassociated with a transmitter and receiver, respectively, operable forinterfacing with external devices.

Additionally, the computer device 700 may include a data store 740,which can be any suitable combination of hardware and/or software, thatprovides for mass storage of information, databases, and programsemployed in connection with implementations described herein. Forexample, the data store 740 may be a data repository for operatingsystem 760 (e.g., classical OS, or quantum OS). In one implementation,the data store 740 may include the memory 720.

The computer device 700 may also include a user interface component 750operable to receive inputs from a user of the computer device 700 andfurther operable to generate outputs for presentation to the user or toprovide to a different system (directly or indirectly). The userinterface component 750 may include one or more input devices, includingbut not limited to a keyboard, a number pad, a mouse, a touch-sensitivedisplay, a digitizer, a navigation key, a function key, a microphone, avoice recognition component, any other mechanism capable of receiving aninput from a user, or any combination thereof. Further, the userinterface component 750 may include one or more output devices,including but not limited to a display, a speaker, a haptic feedbackmechanism, a printer, any other mechanism capable of presenting anoutput to a user, or any combination thereof.

In an implementation, the user interface component 750 may transmitand/or receive messages corresponding to the operation of the operatingsystem 760. In addition, the processor 710 may execute the operatingsystem 760 and/or applications or programs (e.g., programs to generate,identify, and use various implementations of the controlled-Z^(a) gateand/or the Heisenberg interaction), and the memory 720 or the data store740 may store them.

When the computer device 700 is implemented as part of a cloud-basedinfrastructure solution, the user interface component 750 may be used toallow a user of the cloud-based infrastructure solution to remotelyinteract with the computer device 700.

FIG. 8 is a flow diagram that illustrates an example of a method 800 forperforming a quantum algorithm in accordance with aspects of thisdisclosure. In an aspect, the method 800 may be performed in a computersystem such as the computer system 700 described above, where, forexample, the processor 710, the memory 720, the data store 740, and/orthe operating system 760 may be used to perform the functions of themethod 800. In another aspect, the method 800 may be performed in a QIPsystem such as the QIP system 1000 in FIG. 10 . The QIP system may be afault-tolerant QIP system.

At 810, use of a controlled-Z^(a) gate is identified as part of thequantum algorithm, where a is a parameter and a∈[−1, 1]. In somenon-limiting examples, the parameter a can take any number of values,including a=1, a=½, a=−½, a=¼, or a=−¼, depending on the gate that isbeing implemented.

At 820, the controlled-Z^(a) gate is implemented for the fault-tolerantQIP system.

In a first scenario, the implementation of the controlled-Z^(a) gate iswithout or not including measurement and feedforward. In such ascenario, the implementation of the controlled-Z^(a) gate includesmultiple elements with only six (6) of the elements being CNOT gates.The multiple elements of the implementation of the controlled-Z^(a) gatecan further include one (1) parametrized Z^(a) gate, eight (8) T gates,four (4) Hadamard gates, and one (1) ancilla.

In a second scenario, the implementation of the controlled-Z^(a) gate iswith or including measurement and feedforward. In such a scenario, theimplementation of the controlled-Z^(a) gate includes multiple elementswith only four (4) of the elements being T gates, only three (3) of theelements being CNOT gates, and only three (3) of the elements beingHadamard gates. The multiple elements of the implementation of thecontrolled-Z^(a) gate can further include one (1) parametrized Z^(a)gate, one (1) ancilla, one (1) measurement, and one (1)classically-conditioned controlled Pauli-Z gate.

In both the first scenario and the second scenario described above inconnection with 820 of the method 800, the controlled-Z^(a) gate isimplemented without using P gates and P^(†) gates.

At 830, the implementation of the controlled-Z^(a) gate is mapped into aphysical representation in the fault-tolerant QIP system.

In an aspect of the method 800, the fault-tolerant QIP system is atrapped-ion QIP system, and mapping the implementation of thecontrolled-Z^(a) gate includes mapping the implementation of thecontrolled-Z^(a) gate using multiple trapped-ion-based qubits in thetrapped-ion QIP system.

In another aspect of the method 800, the fault-tolerant QIP system is asuperconducting QIP system, and mapping the implementation of thecontrolled-Z^(a) gate includes mapping the implementation of thecontrolled-Z^(a) gate using multiple superconducting-based qubits in thesuperconductor QIP system.

At 840, the quantum algorithm is performed based at least in part on thephysical representation. In an aspect, the quantum algorithm can be aQFT.

FIG. 9 is a flow diagram that illustrates an example of a method forperforming a quantum simulation in accordance with aspects of thisdisclosure. In an aspect, the method 900 may be performed in a computersystem such as the computer system 700 described above, where, forexample, the processor 710, the memory 720, the data store 740, and/orthe operating system 760 may be used to perform the functions of themethod 900. In another aspect, the method 900 may be performed in a QIPsystem such as the QIP system 1000 in FIG. 10 . The QIP system may be afault-tolerant QIP system.

At 910, use of a Heisenberg interaction is identified as part of thequantum simulation.

At 920, a controlled-Z^(a) gate for implementing the Heisenberginteraction is identified, wherein a is a parameter and a∈[−1, 1]. Insome non-limiting examples, the parameter a can take any number ofvalues, including a=1, a=½, a=−½, a=¼, or a=−¼, depending on the gatethat is being implemented.

At 930, the Heisenberg interaction is implemented for the fault-tolerantQIP system, wherein the implementation of the Heisenberg interaction isbased on an implementation of the controlled-Z^(a) gate and includesmultiple elements with only one (1) of the elements being a parametrizedZ^(a) gate.

In a first scenario, the implementation of the Heisenberg interactionincludes identifying that the implementation of the controlled-Z^(a)gate is without measurement and feedforward. In such a scenario, theelements of the implementation of the Heisenberg interaction can furtherinclude eight (8) T gates, eight (8) CNOT gates, six (6) Hadamard gates,and one (1) ancilla.

In a second scenario, the implementation of the Heisenberg interactionincludes identifying that the implementation of the controlled-Z^(a)gate is with measurement and feedforward. In such a scenario, theelements of the implementation of the Heisenberg interaction can furtherinclude four (4) T gates, five (5) CNOT gates, five (5) Hadamard gates,one (1) ancilla, one (1) measurement, and one (1)classically-conditioned controlled Pauli-Z gate.

In both the first scenario and the second scenario described above inconnection with 930 of the method 900, the Heisenberg interaction isimplemented without using P gates and P^(†) gates.

At 940, the implementation of the Heisenberg interaction is mapped intoa physical representation in the fault-tolerant QIP system.

In an aspect of the method 900, the fault-tolerant QIP system is atrapped-ion QIP system, and mapping the implementation of the Heisenberginteraction includes mapping the implementation of the Heisenberginteraction using multiple trapped-ion-based qubits in the trapped-ionQIP system.

In another aspect of the method 900, the fault-tolerant QIP system is asuperconducting QIP system, and mapping the implementation of theHeisenberg interaction includes mapping the implementation of theHeisenberg interaction using multiple superconducting-based qubits inthe superconducting QIP system.

At 950, the quantum simulation is performed based at least in part onthe physical representation. In an aspect, the quantum simulation can bea many body localization phenomena simulation.

At 960, the results from the quantum simulation are provided.

In another method for performing a quantum simulation, the method mayinclude identifying use of a Heisenberg interaction as part of thequantum simulation, implementing a pre-fault tolerant implementation ofthe Heisenberg interaction (see e.g., the diagram 600 in FIG. 6 ), wherethe pre-fault tolerant implementation of the Heisenberg interactionincludes multiple elements with only three (3) of the elements beingCNOT gates, mapping the pre-fault tolerant implementation of theHeisenberg interaction into a physical representation in a QIP system,performing the quantum simulation based at least in part on the physicalrepresentation, providing results from the quantum simulation. Themultiple elements of the pre-fault tolerant implementation of theHeisenberg interaction further include three (3) R_(z) gates, and two(2) R_(y) gates, where the three (3) R_(z) gates include one (1)R_(z)(−π/2), one (1) R_(z)(2a−π/2), and one (1) R_(z)(π/2), the two (2)R_(y) gates include (1) R_(y)(π/2−2a) and one (1) R_(y)(2a−π/2), and ais a parameter and a e [−1, 1]. A QIP may be configured for performingthis quantum simulation and a computer-readable medium may store codewith instructions executable by a processor for performing this quantumsimulation.

FIG. 10 is a block diagram that illustrates an example of a QIP system1000 in accordance with aspects of this disclosure. The QIP system 1000may also be referred to as a quantum computing system, a computerdevice, or the like. In an aspect, the QIP system 1000 may correspond toportions of a quantum computer implementation of the computer device 700in FIG. 7 .

The QIP system 1000 can include a source 1060 that provides atomicspecies to a chamber 1050 having an ion trap 1070 that traps the atomicspecies once ionized by an optical controller 1020. Optical sources 1030in the optical controller 1020 may include one or more laser sourcesthat can be used for ionization of the atomic species, control (e.g.,phase control) of the atomic ions, and for fluorescence of the atomicions that can be monitored and tracked by image processing algorithmsoperating in an imaging system 1040 in the optical controller 1020. Theimaging system 1040 can include a high resolution imager (e.g., CCDcamera) for monitoring the atomic ions while they are being provided tothe ion trap 1070 (e.g., for counting) or after they have been providedto the ion trap 1070 (e.g., for monitoring the atomic ions states). Inan aspect, the imaging system 1040 can be implemented separate from theoptical controller 1020, however, the use of fluorescence to detect,identify, and label atomic ions using image processing algorithms mayneed to be coordinated with the optical controller 1020.

The QIP system 1000 may also include an algorithms component 1010 thatmay operate with other parts of the QIP system 1000 (not shown) toperform quantum algorithms (e.g., QFT, quantum simulations) that makeuse of the implementations described above. An implementation component1015 may be used to identify, generate, select, or otherwise determine aparticular implementation to be used of a controlled-Z^(a) gate and/or aHeisenberg interaction. The implementation component 1015 may operate inconjunction with a circuit optimizer, may it be the circuit optimizer orat least a part of the circuit optimizer, and/or may implement aspectsor functions of a circuit optimizer. In an aspect, the implementationcomponent 1015 may operate with the algorithms component 1010 to breakdown code for quantum computations or quantum simulations into computingor gate primitives that can be physically implemented. As such, thealgorithms component 1010 may provide instructions to various componentsof the QIP system 1000 (e.g., to the optical controller 1020) to enablethe implementation of quantum circuits, or their equivalents, such asthe ones described herein. That is, the algorithms component 1010 mayallow for mapping of different computing primitives into physicalrepresentations using, for example, the ion trap 1070. It is to beunderstood that the QIP system 1000, the algorithms component 1010,and/or the implementation component 1015 may operate in a fault-tolerantmode or setting in connection with the implementation of acontrolled-Z^(a) gate and/or a Heisenberg interaction.

As described herein, either or both of the computer device 700 or theQIP system 1000 may be configured as fault-tolerant quantum computingsystems. Moreover, while the QIP system 1000 has been described as beingan ion-trapped-based system, similar operations/functionality may beachieved with a superconducting-based system.

Although the present disclosure has been provided in accordance with theimplementations shown, one of ordinary skill in the art will readilyrecognize that there could be variations to the embodiments and thosevariations would be within the scope of the present disclosure.Accordingly, many modifications may be made by one of ordinary skill inthe art without departing from the scope of the appended claims.

What is claimed is:
 1. A method for performing a quantum algorithm,comprising: identifying use of a controlled-Z^(a) gate as part of thequantum algorithm, wherein the quantum algorithm includes a Heisenberginteraction and is based on projecting a real-valued degree of freedomin the Heisenberg interaction onto a controlled R_(z) ^(a) rotation, andwherein a is a parameter and a ∈[−1, 1]; implementing thecontrolled-Z^(a) gate with an ancilla qubit for a fault-tolerant quantuminformation processing (QIP) system, wherein the implementation of thecontrolled-Z^(a) gate includes multiple elements, and wherein themultiple elements include a single parametrized Z^(a) gate, only four(4) Hadamard gates, only four (4) T gates, only four (4) T^(†) gates,and only six (6) controlled-NOT (CNOT) gates; mapping the implementationof the controlled-Z^(a) gate into a physical representation in thefault-tolerant QIP system; and performing the quantum algorithm based atleast in part on the physical representation.
 2. The method of claim 1,wherein the quantum algorithm is a quantum Fourier transform (QFT). 3.The method of claim 1, wherein implementing the controlled-Z^(a) gatefor the fault-tolerant QIP system includes identifying that theimplementation of the controlled-Z^(a) gate is without measurement andfeedforward.
 4. The method of claim 1, wherein: the fault-tolerant QIPsystem is a trapped-ion QIP system, and mapping the implementation ofthe controlled-Z^(a) gate includes mapping the implementation of thecontrolled-Z^(a) gate using multiple trapped-ion-based qubits in thetrapped-ion QIP system.
 5. The method of claim 1, wherein: thefault-tolerant QIP system is a superconducting QIP system, and mappingthe implementation of the controlled-Z^(a) gate includes mapping theimplementation of the controlled-Z^(a) gate using multiplesuperconducting-based qubits in the superconducting QIP system.
 6. Afault-tolerant quantum information processing (QIP) system forperforming a quantum algorithm, comprising: an implementation componentconfigured to: identify use of a controlled-Z^(a) gate as part of thequantum algorithm, wherein the quantum algorithm includes a Heisenberginteraction and is based on projecting a real-valued degree of freedomin the Heisenberg interaction onto a controlled R_(z) ^(a) rotation,wherein a is a parameter and a∈[−1, 1], implement the controlled-Z^(a)gate with an ancilla qubit for the fault-tolerant QIP system, whereinthe implementation of the controlled-Z^(a) gate includes multipleelements, and wherein the multiple elements include a singleparametrized Z^(a) gate, only four (4) Hadamard gates, only four (4) Tgates, only four (4) T^(†) gates, and only six (6) controlled-NOT (CNOT)gates, and map the implementation of the controlled-Z^(a) gate into aphysical representation in the fault-tolerant QIP system map; and analgorithms component configured to perform the quantum algorithm basedat least in part on the physical representation.
 7. A non-transitorycomputer-readable medium storing code with instructions executable by aprocessor for performing a quantum algorithm, wherein the quantumalgorithm includes a Heisenberg interaction and is based on projecting areal-valued degree of freedom in the Heisenberg interaction onto acontrolled R_(z) ^(a) rotation, comprising: code for identifying use ofa controlled-Z^(a) gate as part of the quantum algorithm, wherein a is aparameter and a∈[−1, 1]; code for implementing the controlled-Z^(a) gatewith an ancilla qubit for a fault-tolerant quantum informationprocessing (QIP) system, wherein the implementation of thecontrolled-Z^(a) gate includes multiple elements, and wherein themultiple elements include a single parametrized Z^(a) gate, only four(4) Hadamard gates, only four (4) T gates, only four (4) T^(†) gates,and only six (6) controlled-NOT (CNOT) gates; code for mapping theimplementation of the controlled-Z^(a) gate into a physicalrepresentation in the fault-tolerant QIP system; and code for performingthe quantum algorithm based at least in part on the physicalrepresentation.
 8. A method for performing a quantum algorithm,comprising: identifying use of a controlled-Z^(a) gate as part of thequantum algorithm, wherein the quantum algorithm includes a Heisenberginteraction and is based on projecting a real-valued degree of freedomin the Heisenberg interaction onto a controlled R_(z) ^(a) rotation,wherein a is a parameter and a∈[−1, 1]; implementing the controlled-Z″gate with an ancilla qubit for a fault-tolerant quantum informationprocessing (QIP) system, wherein the implementation of the controlled-Z′gate includes multiple elements, and the multiple elements include asingle parametrized Z′ gate, only two (2) T gates, only two (2) T^(†)gates only three (3) controlled-NOT (CNOT) gates, and only three (3)Hadamard (H) gates; mapping the implementation of the controlled-Za gateinto a physical representation in the fault-tolerant QIP system; andperforming the quantum algorithm based at least in part on the physicalrepresentation.
 9. The method of claim 8, wherein the quantum algorithmis a quantum Fourier transform (QFT).
 10. The method of claim 8, whereinimplementing the controlled-Z^(a) gate for the fault-tolerant QIP systemincludes identifying that the implementation of the controlled-Z^(a)gate is with measurement and feedforward.
 11. The method of claim 8,wherein the multiple elements of the implementation of thecontrolled-Z^(a) gate further include: a single measurement element, anda single classically-conditioned controlled Pauli-Z gate.
 12. The methodof claim 8, wherein: the fault-tolerant QIP system is a trapped-ion QIPsystem, and mapping the implementation of the controlled-Z^(a) gateincludes mapping the implementation of the controlled-Z^(a) gate usingmultiple trapped-ion-based qubits in the trapped-ion QIP system.
 13. Themethod of claim 8, wherein: the fault-tolerant QIP system is asuperconducting QIP system, and mapping the implementation of thecontrolled-Z^(a) gate includes mapping the implementation of thecontrolled-Z^(a) gate using multiple superconducting-based qubits in thesuperconducting QIP system.
 14. A fault-tolerant quantum informationprocessing (QIP) system for performing a quantum algorithm, comprising:an implementation component configured to: identify use of acontrolled-Z^(a) gate as part of the quantum algorithm, wherein thequantum algorithm includes a Heisenberg interaction and is based onprojecting a real-valued degree of freedom in the Heisenberg interactiononto a controlled R_(z) ^(a) rotation, wherein a is a parameter anda∈[−1, 1], implement the controlled-Z^(a) gate with an ancilla qubit forthe fault-tolerant QIP system, wherein the implementation of thecontrolled-Za gate includes multiple elements, and wherein the multipleelements include a single parametrized Z^(a) gate, only two (2) T gates,only two (2) T^(†) gates, only three (3) controlled-NOT (CNOT) gates,and only three (3) Hadamard (H) gates, and map the implementation of thecontrolled-Z^(a) gate into a physical representation in thefault-tolerant QIP system; and an algorithms component configured toperform the quantum algorithm based at least in part on the physicalrepresentation.
 15. A non-transitory computer-readable medium storingcode with instructions executable by a processor for performing aquantum algorithm, comprising: code for identifying use of acontrolled-Z^(a) gate as part of the quantum algorithm, wherein thequantum algorithm includes a Heisenberg interaction and is based onprojecting a real-valued degree of freedom in the Heisenberg interactiononto a controlled R_(z) ^(a) rotation, wherein a is a parameter anda†[−1, 1]; code for implementing the controlled-Z^(a) gate with anancilla qubit for a fault-tolerant quantum information processing (QIP)system, wherein the implementation of the controlled-Z^(a) gate includesmultiple elements, and wherein the multiple elements include a singleparametrized Z^(a) gate, only two (2) T gates, only two (2) T^(†) gates,only three (3) controlled-NOT (CNOT) gates, and only three (3) Hadamard(H) gates; code for mapping the controlled-Z^(a) gate into a physicalrepresentation in the fault-tolerant QIP system; and code for performingthe quantum algorithm based at least in part on the physicalrepresentation.
 16. A method for performing a quantum simulation,comprising: identifying use of a Heisenberg interaction as part of thequantum simulation; identifying a controlled-Z^(a) gate for implementingthe Heisenberg interaction by projecting a real-valued degree of freedomin the Heisenberg interaction onto a controlled R_(z) ^(a) rotation,wherein a is a parameter and a∈[−1, 1]; implementing the Heisenberginteraction for a fault-tolerant quantum information processing (QIP)system, wherein the implementation of the Heisenberg interaction isbased on an implementation of the controlled-Z^(a) gate with an ancillaqubit, and wherein the implementation of the Heisenberg interactionincludes a single parametrized Z^(a) gate, only six (6) Hadamard gates,only four (4) T gates, only four (4) T^(†) gates, and only eight (8)controlled-NOT (CNOT) gates; mapping the implementation of theHeisenberg interaction into a physical representation in thefault-tolerant QIP system; performing the quantum simulation based atleast in part on the physical representation; and providing results fromthe quantum simulation.
 17. The method of claim 16, wherein the quantumsimulation is configured for solving a problem associated with a manybody localization phenomena.
 18. The method of claim 16, whereinimplementing the Heisenberg interaction for the fault-tolerant QIPsystem includes identifying that the implementation of thecontrolled-Z^(a) gate is without measurement and feedforward.
 19. Themethod of claim 16, wherein implementing the Heisenberg interaction forthe fault-tolerant QIP system includes identifying that theimplementation of the controlled-Z^(a) gate is with measurement andfeedforward.
 20. The method of claim 16, wherein: the fault-tolerant QIPsystem is a trapped-ion QIP system, and mapping the implementation ofthe Heisenberg interaction includes mapping the implementation of theHeisenberg interaction using multiple trapped-ion-based qubits in thetrapped-ion QIP system.
 21. The method of claim 16, wherein: thefault-tolerant QIP system is a superconducting QIP system, and mappingthe implementation of the Heisenberg interaction includes mapping theimplementation of the Heisenberg interaction using multiplesuperconducting-based qubits in the superconducting QIP system.
 22. Afault-tolerant quantum information processing (QIP) system forperforming a quantum simulation, comprising: an implementation componentconfigured to: identify use of a Heisenberg interaction as part of thequantum simulation, identify a controlled-Z^(a) gate for implementingthe Heisenberg interaction by projecting a real-valued degree of freedomin the Heisenberg interaction onto a controlled R_(z) ^(a) rotation,wherein a is a parameter and a∈[−1, 1], implement the Heisenberginteraction for the fault-tolerant QIP system, wherein theimplementation of the Heisenberg interaction is based on animplementation of the controlled-Z^(a) gate with an ancilla qubit, andwherein the implementation of the Heisenberg interaction includes asingle parametrized Z^(a) gate, only six (6) Hadamard gates, only four(4) T gates, only four (4) T^(†) gates, and only eight (8)controlled-NOT (CNOT) gates, map the implementation of the Heisenberginteraction into a physical representation in the fault-tolerant QIPsystem; and an algorithms component configured to: perform the quantumalgorithm based at least in part on the physical representation, andprovide results from the quantum simulation.
 23. A non-transitorycomputer-readable medium storing code with instructions executable by aprocessor for performing a quantum simulation, comprising: code foridentifying use of a Heisenberg interaction as part of the quantumsimulation; code for identifying a controlled-Z^(a) gate forimplementing the Heisenberg interaction by projecting a real-valueddegree of freedom in the Heisenberg interaction onto a controlled R_(z)^(a) rotation, wherein a is a parameter and a∈[−1, 1]; code forimplementing the Heisenberg interaction for a fault-tolerant quantuminformation processing (QIP) system, wherein the implementation of theHeisenberg interaction is based on an implementation of thecontrolled-Z^(a) gate with an ancilla qubit, and wherein theimplementation of the Heisenberg interaction includes a singleparametrized Z^(a) gate, only six (6) Hadamard gates, only four (4) Tgates, only four (4) T^(†) gates, and only eight (8) controlled-NOT(CNOT) gates; code for mapping the implementation of the Heisenberginteraction into a physical representation in the fault-tolerant QIPsystem; code for performing the quantum simulation based at least inpart on the physical representation; and code for providing results fromthe quantum simulation.
 24. A method for performing a quantumsimulation, comprising: identifying use of a Heisenberg interaction aspart of the quantum simulation; identifying a controlled-Z^(a) gate forimplementing the Heisenberg interaction by projecting a real-valueddegree of freedom in the Heisenberg interaction onto a controlled R_(z)^(a) rotation, wherein a is a parameter and a∈[−1, 1]; implementing theHeisenberg interaction for a fault-tolerant quantum informationprocessing (QIP) system, wherein the implementation of the Heisenberginteraction is based on an implementation of the controlled-Z^(a) gatewith an ancilla qubit, and wherein the implementation of the Heisenberginteraction includes a single parametrized Z^(a) gate, only five (5)Hadamard gates, only two (2) T gates, only two (2) T^(†) gates, and onlyfive (5) controlled-NOT (CNOT) gates; mapping the implementation of theHeisenberg interaction into a physical representation in thefault-tolerant QIP system; performing the quantum simulation based atleast in part on the physical representation; and providing results fromthe quantum simulation.
 25. The method of claim 24, wherein theimplementation of the Heisenberg interaction further includes: a singlemeasurement element, and a single classically-conditioned controlledPauli-Z gate.